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Consider a free scalar field theory. My struggle is that vacuum correlation functions of fields are only Lorentz invariant under a subgroup of Lorentz transformations, despite the invariance of the vacuum under the complete group of Lorentz transformations! I expect that I am making suspect assumptions somewhere.


I expect the vacuum to be invariant under more than just proper, orthochronous Lorentz transformations: I expect the vacuum to be invariant under time reversal invariance and spatial inversion: $T|0\rangle = |0\rangle$ and $P|0\rangle = |0\rangle$, where these operators act on the field operators as $T^{-1} \psi(x) T = \psi(\Lambda_Tx)$ and $P^{-1} \psi(x) P = \psi(\Lambda_Px)$ where $\Lambda_T$ and $\Lambda_P$ are the usual 4x4 matrices for time reversal and inversion.

However, vacuum invariance implies invariance of correlation functions: consider \begin{align*} D(x,y) &= \langle 0| \psi(x) \psi(y) |0\rangle \\ &= \langle 0|P^{-1}P \psi(x) P^{-1}P \psi(y) P^{-1}P|0\rangle \\ &= \langle 0|\psi(\Lambda_Px)\psi(\Lambda_Py)|0\rangle \\ &= D(\Lambda_Px, \Lambda_Py) \end{align*}

This holds similarly for $T$, $D(x,y) = D(\Lambda_Tx,\Lambda_Ty)$.

However, (see below) I don't think $D(x,y) = D(\Lambda_Tx,\Lambda_Ty)$ is true!

The fact that $D(x,y) = \langle 0| \psi(x) \psi(y) |0\rangle$ is only invariant ($D(\Lambda x, \Lambda y) = D(x,y)$) under proper, orthochronous Lorentz transformations and not generic Lorentz transformations comes up in discussing causality. Invariance under proper, orthochronous transformations means the commutator $[\psi(x),\psi(y)]$ will vanish for spacelike $x-y$, which it does. Invariance under all transformations would mean the commutator would vanish for timelike $x-y$, but it doesn't! See also A question about causality and Quantum Field Theory from improper Lorentz transformation for background.


What am I getting wrong?

My guesses for what's wrong above:

  1. The vacuum is not invariant under time reversal and spatial inversion. Seems unlikely to me.
  2. The fields transform differently under the operator implementations of $T$ and $P$. Seems unlikely to me.
  3. My insertions of $I = P^{-1} P$ and $I = T^{-1} T$ are mistaken, perhaps in the latter case by the anti-unitarity of the operator implementation of $T$. Unsure.
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Are $C$, $P$ symmetries of your model individually?

If yes, then no wonder your correlation functions are invariant under them.

If no, there doesn’t exist a unitary operator $P$ that acts on fields in the way you described. In fact, $P$ will act as

$$ \psi \rightarrow P \psi P^{\dagger}, $$

which when $P$ isn’t unitary doesn’t cancel like you expect it to.

W.r.t. $T$ – because it is anti-linear, the story is a bit more involved. Unlike with unitary symmetries, anti-unitary symmetries actually don't preserve inner products – they only preserve probabilities. Hence, the correlation function, which is expressed as an inner product, can and will change under time reversal. Its absolute value squared, however, will not (for $T$-invariant models; for models with $T$ violation, which is the same as $CP$ violation due to the $CPT$ theorem, it will).

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  • $\begingroup$ Thank you. In my particular case, yes, I have $P$ and $T$ invariance of the theory because I had in mind a quadratic scalar theory, so in my case it must boil down to $T$ only preserving absolute values. This makes sense given that in my case $D(x,y)$ is real and nonnegative for spacelike $x-y$ but not necessarily real for timelike $x-y$, showing that $D(x,y) = D(\Lambda_Tx,\Lambda_Ty)$ holds only spacelike $x-y$. $\endgroup$
    – user196574
    Commented Nov 2, 2020 at 20:18

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